Kanamori-Moir\'e-Hubbard model for transition metal dichalcogenide homobilayers

Ab-initio and continuum model studies predicted that the $\Gamma$ valley transition metal dichalcogenide (TMD) homobilayers could simulate the conventional multi-orbital Hubbard model on the moir\'e Honeycomb lattice. Here, we perform the Wannierization starting from the continuum model and show that a more general Kanamori-Moir\'e-Hubbard model emerges, beyond the extensively studied standard multi-orbital Hubbard model, which can be used to investigate the many-body physics in the $\Gamma$ valley TMD homobilayers. Using the unrestricted Hartree-Fock and Lanczos techniques, we study these half-filled multi-orbital moir\'e bands. By constructing the phase diagrams we predict the presence of an antiferromagnetic state and in addition we found unexepected and dominant states, such as a S = 1 ferromagnetic insulator and a charge density wave state. Our theoretical predictions made using this model can be tested in future experiments on the $\Gamma$ valley TMD homobilayers.